Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Bréhier, Charles-Edouard

The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 175-228

We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $ϵ$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $\Delta t$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $ϵ$, in terms of $\Delta t$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of the recent article [10] in an infinite dimensional framework. The fast component is discretized using the modified Euler scheme for SPDEs introduced in the recent work [8]. Proving the weak error estimates requires delicate analysis of the regularity properties of solutions of infinite dimensional Kolmogorov equations. Numerical experiments illustrate the asymptotic preserving property and the uniform weak error estimates.

Classification : 60H35, 65C30, 60H15
Keywords: Stochastic partial differential equations, asymptotic preserving schemes, Euler schemes, infinite dimensional Kolmogorov equations

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